Theorem nnssn0s 28317

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28311	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4088	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3980	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3898   ⊆ wss 3901  {csn 4580   0_s c0s 27801  ℕ_0scn0s 28308  ℕ_scnns 28309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-ss 3918  df-nns 28311

This theorem is referenced by:  nnssno  28318  nnn0s  28323  nnn0sd  28324  nnsgt0  28335